The simplest application of the formalism, as well as a consistency check, is the flat spacetime limit. In flat space eq.(REF ), reduces to the QFT formula derived in ref. [1]}.
Indeed, in this case, we can choose the Cauchy hypersurfaces to be the \(t = constant\)  surfaces in a given Minkowskian coordinate system, then the modes \(\lbrace \zeta _{k,i} (x), \zeta ^{*}_{k,i}(x) \rbrace \)  coincide with the plane wave solutions of the flat Klein-Gordon equation, so that \(\Lambda _{q;k} (t) \rightarrow U_{q;k}(t)\)  and \(\Xi _{q;k}(t) \rightarrow V_{q,r;k,s} (t)\) . Here \(U_{q;k}\)  and \(V_{q;k}\)  are the Bogoliubov coefficients for boson mixing in flat space: \(U_{q;k}=\delta ^{3}(k-q)|U_{k}|\cdot e^{i(\omega _{k,2}-\omega _{k,1})t}\)  and \(V_{q;k}=\delta ^{3}(k-q)|V_{k}|\cdot e^{i(\omega _{k,1}+\omega _{k,2})t}\) . Here the label \(k\)  stands for the three-momentum \(k\)  and \(\omega _{k,i} = \sqrt{k^2 + m_i^2}\) .
The transition probability of eq.(REF ) is thus given by
\(\begin{split}P_{A \rightarrow B}(t) &=2 \cos ^{2}(\theta )\sin ^{2}(\theta )(1-\Re [U^{*}_{k}(0) U_{k}(t)-V^{*}_{k}(0) V_{k}(t)]) \\&=2 \cos ^{2}(\theta ) \sin ^{2}(\theta )\bigg [1-\Re \left[|U_{k}(0)|^2 e^{i(\omega _{k,2} - \omega _{k,1})t} - |V_{k}(0)|^2e^{i(\omega _{k,2} + \omega _{k,1})t}\right]\bigg ]\\&=2 \cos ^{2}(\theta ) \sin ^{2}(\theta )\bigg [1- |U_{k}(0)|^2 \cos [(\omega _{k,2} - \omega _{k,1})t] + |V_{k}(0)|^2 \cos [(\omega _{k,2} + \omega _{k,1})t]\bigg ]\\&=\sin ^{2}(2\theta )\bigg [|U_{k}(0)|^2 \left(\frac{1}{2}-\cos [(\omega _{k,2} - \omega _{k,1})t] \right) +|V_{k}(0)|^2 \left(-\frac{1}{2}+\cos [(\omega _{k,2} + \omega _{k,1})t] \right)\bigg ]\,,\end{split}\) 
